# Fibonacci Series and the shape of the Milky Way **URL:** \ **Category:** Factual Questions\ **Created:** [February 26, 2004, 2:52pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289 "2004-02-26T14:52:53Z")\ **Posts on this page:** 15\ **Page:** 1
**Author:** ![Achilles](https://avatars.discourse-cdn.com/v4/letter/a/dc4da7/32.png) [@Achilles](https://boards.straightdope.com/u/Achilles)\ **Post date:** [February 26, 2004, 2:52pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/1 "2004-02-26T14:52:53Z")
Why would the ratio 1.61…blahblahblah be reflected in the growth pattern of galaxies ? Our maths lecturer made it seem that _all_ spiral galaxies adhere to the Golden Ratio. Is this true, and if so, what is so fundamental about _this_ particular irrational number that would enable it to be reflected in so many places (plants, shellfish et cetera…) ---
**Author:** ![rsa](https://avatars.discourse-cdn.com/v4/letter/r/839c29/32.png) [@rsa](https://boards.straightdope.com/u/rsa)\ **Post date:** [February 26, 2004, 6:17pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/2 "2004-02-26T18:17:34Z")
Check out this [space.com](http://space.com) article. [Spirals in Nature: The Magical Number behind Hurricanes and Galaxies](http://www.space.com/scienceastronomy/perfect_spirals_030917.html) ---
**Author:** ![Rico](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rico/32/3582_2.png) [@Rico](https://boards.straightdope.com/u/Rico)\ **Post date:** [February 26, 2004, 6:35pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/3 "2004-02-26T18:35:52Z")
And, if I rememebr correctly from all those years ago, the Golden Ratio (which you’re talking about) has nothing to do with the Fibonacci series (1,1,2,3,5,8,13,21…) where the next number in the series is the sum of the two preceding numbers. Am I right? ---
**Author:** ![raygirvan](https://avatars.discourse-cdn.com/v4/letter/r/ac91a4/32.png) [@raygirvan](https://boards.straightdope.com/u/raygirvan)\ **Post date:** [February 26, 2004, 7:02pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/4 "2004-02-26T19:02:16Z")
> [@Rico](#): > > Am I right? No. It’s the limiting value approached by the ratio of successive terms in a Fibonacci series. ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [February 26, 2004, 8:16pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/5 "2004-02-26T20:16:19Z")
> [@raygirvan](#): > > No. It’s the limiting value approached by the ratio of successive terms in a Fibonacci series. I find it fascinating that this is true, while at the same time, it can be directly derived from something like: Take a rectangle with sides of length x and y, with x being the longer side. These sides have a ratio x/y (longer over shorter). Add a right-angled side inside the rectangle such that a square is created within and entirely to one end/side of the rectangle. If the original x and y are chosen correctly such that the small rectangle attached to the square has the sides in the same ratio as the original rectangle, then that ratio is the Golden Ratio. (The new sides are, of course, y and x-y, so the ratio is y/(x-y) for the little rectangle). I’m sure there’s a very very long and complicated reason why the Fibonacci series ratio limits approach this value, but just looking at it as a layman, you have to go, ‘whoa… that’s weird’. 😃 Of course, you can do the same process to the little rectangle as well, resulting in an even smaller rectangle, ad infinitum, all with Golden Ratio relationships. If you do this in a sort of ‘circular’ pattern, and connect the centers (let’s say) of the squares, you get a neat spiral… A golden spiral? =) It’s a fascinating number that shows up in wild places, like the Fibonacci series does… ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [February 26, 2004, 8:28pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/6 "2004-02-26T20:28:40Z")
[The Logarithmic Spiral](http://mathworld.wolfram.com/LogarithmicSpiral.html). That page goes into a small bit of detail as to how they’re constructed. It’s sort of like the polar equivalent of a diagonal line. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 26, 2004, 8:33pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/7 "2004-02-26T20:33:42Z")
> [@](#): > > I’m sure there’s a very very long and complicated reason why the Fibonacci series ratio limits approach this value, but just looking at it as a layman, you have to go, ‘whoa… that’s weird’. There’s also a simple and beautiful way to see this. You’re already familiar with the spiral you can form from the Golden Ratio. Now try something similar, but with all squares. Start with a square of length 1. Sick another square next to it: This one must also have length 1. Now, stick a square next to the previous two: It must have length 2. Now another, next to the second and third squares, of length 3. Then one next to the 2 square and the 3 square, so it has a length of 5. The next one is 8, then 13, and so on. Go ahead and draw it out. You should be able to see the pattern in the side lengths, and you should also be able to see the resemblence between the Fibbonocci spiral and the Golden Ratio spiral. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [February 26, 2004, 11:48pm UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/8 "2004-02-26T23:48:44Z")
Here’s another way to show that the sequence of ratios in the Fibonacci sequence converges to the golden ratio. Let a[sub]n[/sub] denote the nth Fibonacci number. Let x be the limit of the ratio a[sub]n+1[/sub] / a[sub]n[/sub]. Then x must also be the limit of the ratio a[sub]n+2[/sub] / a[sub]n+1[/sub] (since this is the exact same ratio, I’ve only changed the indices). Note that: a[sub]n+2[/sub] / a[sub]n+1[/sub] = (a[sub]n+1[/sub] + a[sub]n[/sub]) / a[sub]n+1[/sub] = 1 + a[sub]n[/sub]/a[sub]n+1[/sub] Taking the limit of both sides gives x = 1 + 1/x. Solve for x. One of the two solutions is negative, so we can rule out that possibility; the other value is the golden ratio. Note, however, that we have _not_ shown that the ratio converges to the golden ratio. All I’ve shown so far is that **if** the ratios converge, **then** it must converge to the golden ratio. Without giving it too much thought, I believe we can show the ratios converge by showing that it’s a decreasing sequence bounded below, and we’re done. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [February 27, 2004, 12:00am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/9 "2004-02-27T00:00:11Z")
> [@Cabbage](#): > > Note, however, that we have _not_ shown that the ratio converges to the golden ratio. All I’ve shown so far is that **if** the ratios converge, **then** it must converge to the golden ratio. Without giving it too much thought, I believe we can show the ratios converge by showing that it’s a decreasing sequence bounded below, and we’re done. Well good luck with _that_. (It’s not a decreasing sequence; it’s an oscillating sequence.) ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [February 27, 2004, 12:50am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/10 "2004-02-27T00:50:35Z")
Thanks for pointing that out, **Achernar** , I gotta admit I was just mentioning the possibility off the top of my head without bothering to check if it was actually true. ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [February 27, 2004, 2:01am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/11 "2004-02-27T02:01:57Z")
First: Full props to the math gurus of this board. All honor and glory is yours. That being said, the article linked seemed to say that any logarithmic spiral was related to phi. I thought a logarithmic spiral was anything of the form r = e[sup]theta[/sup] and thus has a constant angle of tangency to a radial line. Am I wrong here? Spiral galaxies have wildly differing degrees of twist to their arms and I can’t see how they’re all related to phi. For the full explanation of the magic ratio in nature buy my upcoming book: _Lego Blocks and the Golden Ratio in Nature_ (Putznam Press 2004). ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [February 27, 2004, 6:25am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/12 "2004-02-27T06:25:51Z")
Did you know that if you take the 34th, 55th, 89th, etc letters from the posts in this thread, it spells HIYRUU? 😃 ---
**Author:** ![Rico](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rico/32/3582_2.png) [@Rico](https://boards.straightdope.com/u/Rico)\ **Post date:** [February 27, 2004, 6:51am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/13 "2004-02-27T06:51:47Z")
Yes, but what happens if you arrange those letters in a perfect spiral? 😃 ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [February 27, 2004, 8:25am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/14 "2004-02-27T08:25:33Z")
> [@MonkeyMensch](#): > > First: Full props to the math gurus of this board. All honor and glory is yours. > > That being said, the article linked seemed to say that any logarithmic spiral was related to phi. I thought a logarithmic spiral was anything of the form r = e[sup]theta[/sup] and thus has a constant angle of tangency to a radial line. Well, in a logarithmic spiral with equation r = exp(b theta), a rotation by 90deg is the same as a scaling by a factor of exp(b pi/2). So clearly, if b = ln(2phi/pi), then this factor is exactly equal to phi!!! Actually… I think you’re right. While the “winding square” log spiral with the above expansion factor may show up in plants and seashells and stuff like that, I don’t think it’s the case with galaxies. The Hubble classification system has a character to denote the tightness of spirals. It goes S0, Sa, Sb, Sc, with Sc being the loosest. So, I guess the OP’s lecturer was mistaken. And in the linked article, nobody other than Mario Livio seems to be claiming phi shows up in galactic dynamics. So maybe he’s wrong too… Anyone else? ---
**Author:** ![Achilles](https://avatars.discourse-cdn.com/v4/letter/a/dc4da7/32.png) [@Achilles](https://boards.straightdope.com/u/Achilles)\ **Post date:** [February 27, 2004, 10:47am UTC](https://boards.straightdope.com/t/fibonacci-series-and-the-shape-of-the-milky-way/231289/15 "2004-02-27T10:47:43Z")
Thanks guys - that’s given me a lot of links to follow. Much appreciated.